# Entropy of a refinement of a partition

We consider a probability space $$(X, B, \mu)$$. Let $$\alpha$$ and $$\beta$$ be countable partitions of X. We suppose $$\beta$$ is a refinement of $$\alpha$$, ie that every set in $$\alpha$$ is a union of sets in $$\beta$$. I am interested into the difference in entropy induced by the partitions $$\alpha$$ and $$\beta$$. We call $$\cal{H}$$ the entropy function.

Then, we know:

$${\cal{H}}(\alpha, X) \leq {\cal{H}}(\beta, X)$$.

Could we have an evaluation of their difference?

• What do you mean here by "an evaluation"? Oct 7, 2020 at 14:14

There is a general formula $$H(\alpha \vee \beta) = H(\alpha)+H(\beta|\alpha) .$$ In your case $$\alpha\vee\beta=\beta$$, whence $$H(\beta)-H(\alpha)$$ coincides with the conditional entropy $$H(\beta|\alpha)$$.